Let's investigate 1

Welcome back mathematicians.

We thought today that we would dig into and investigate one of the strategies that we shared yesterday. And we're thinking about the one that was shared by the green team, where they said they could think about working out 23 minus 19 by rethinking the problem as 24 minus 20 and they said they would do this because they then know that the difference between 20 and 24 is 4. And we drew a number line then, that looked like this, where we said 24 would be here, with an arrow to show that my number line continues, and zero would be there. And they did a big jump to say, well then I can just get rid of 20, and I know that that leaves four left.

So I wanted to talk about this idea and spend some time investigating, well how does this strategy work? And in fact we're using a few different strategies at the same time, but what we're essentially doing is this thing of keeping a constant difference.

So let's look at how that works. Because here is 23 and here is 24 and as you can see that's not the same quantity, which is why it looks a little bit weird to get started with, and in fact 24 is one bigger than 23. Now what I'm going to do is just change my representations around a little bit so that we're looking at the same colours. So I've recreated 23. So that my 10s are in blue and green and my ones are in orange, and I've also recreated 24.

So I'm going to take off these cubes here, opps and I need that orange one. And I can have 24 and that makes it a little bit clearer for me to see. And the other thing that I'm about to model, which we don't usually do, when we are doing subtraction, but we are in this case 'cause it will help us, is model the amount that we're taking away, which is 19, I think.

Yes, and I can check it's 19 'cause theres 1 ten and this must be 9 'cause it's one less than 10 here. So what I've done is represent the tens in blue and green and the ones are represented in orange. And if I move this tower into the middle and I'll align them carefully at the bottom.

I can see here that there's a difference of four, so 23 minus 19 is in fact 4, it leaves a difference of 4. But over here, where I've got 24, it actually leaves a difference of five. So how does this work? Well actually, what happened is these guys rethought the numbers and they said well if I increase this number by one. I get 24 and, which means, I also have to increase this quantity by one. And now I'll keep a constant difference of four. Can you see that?

Yes so. That's right, so what they did was they added onto, 1 onto one number and added one onto the other number as well, and that kept the same difference.

Yeah, and so you're right, I could actually add on two to each number. And if I increase each by two. I still keep a difference of 4. So actually over here I also know that 23 is equivalent. 23 minus 19 is equivalent to 24 minus 20 and now I can actually also see it's also equivalent to 25 minus 21. Yeah, I know, and it, well, I could even go crazier if I wanted and I could say, well, I don't want to increase them by blocks. I want to increase them by bananas and if I increase, and precariously balance, this collection on a banana. At the moment, they're not the same, but if I also balance this one on a banana, very carefully, I still have a difference of four. Yes, so, so actually is the same as 23 plus one banana minus, yes 19 plus 1 banana.

Yeah, and so I think you're seeing this, that what really matters here is, is keeping, is the constant difference. So whatever you do to one quantity. You have to do the same to the other. Yeah, and so when the, when the green team was thinking about reworking 23 minus 19 what they were thinking about, where, was this number here. They were like, well, actually, this would be better for me if it were a landmark number. So if I get my 19. And I increase it by one. I get to 20, which is a nicer number for my brain to work with, 'cause it's a multiple of ten. It is a landmark number, which means I also need to increase this number by one to keep a constant difference.

And so you're right, they could also have thought, well, maybe it's not about 19, that I'm worried about, but I could increase 23 by 7 to get a landmark number. So if I increase this collection 23 by 7 more. I end up with 30 or 3 tens. Yeah, and so this distance is no longer 4, no longer 4, so they have to add 7 more, increase it by 7. And that will also leave a

difference of 4. Four, yeah, so in this case we've said that's also the same as 30 minus 19 plus 7 more, which is 26 which also equals 4.

And here's the other thing that's really cool about this is, you don't have to add. You can also subtract. So if I'm starting with 23, actually the closest landmark number would be 20. So if I remove 3 from this number and remove 3 from this number, I still have a difference of 4, but in this case I'm saying 23 minus 19 is equivalent in value to 20 minus 16 and that is how you can work with constant difference.

OK mathematicians, what was the mathematics? So what we realised today is that when we're subtracting, one strategy that we can use to solve the problem is to adjust both of the numbers, so we keep a constant difference. And we saw that today when we saw 23 minus 19 is equivalent in value to 24 minus 20, 25 minus 21, 20 minus 16 and even 23 and one banana minus 19 and one banana.

So over to you now mathematicians to have a go at using this strategy and see how it works for you!

Until next time.